Chiral Nelson--Barr Models: Quality and Cosmology

It was recently shown that domain walls from the spontaneous breaking of CP symmetry are exactly stable, and must be inflated away to recover a viable cosmology. We investigate the phenomenological implications of this result in Nelson--Barr solutions of the strong CP problem. Combined with the upper bound on the scale of spontaneous CP breaking necessary to suppress contributions from dangerous, nonrenormalizable operators to $\bar{\theta}$, this puts an upper bound on the scale of inflation and the reheating temperature after inflation. Minimal Nelson--Barr models are therefore in tension with thermal leptogenesis, models of large-field inflation, or potential future observations of signals from topological remnants of an unrelated, subsequent phase transition. We study how extending Nelson--Barr models with a new, continuous chiral gauge symmetry can ameliorate this tension by forbidding the dangerous dimension-five operators. In particular, we show that gauging a linear combination of baryon number and hypercharge allows for an economic, anomaly-free extension of the minimal Nelson--Barr model, and discuss the phenomenological implications.